Understanding of the intimate cross coupling between electric and magnetic degrees of freedom in solids usually requires sophisticated models and time-consuming calculation methods. Instead of macroscopic symmetry analysis, we present a simple but general approach to explore the microscopic mechanism of magnetoelectric (ME) effects in magnetic ordered materials based on local spin and lattice symmetry analysis. Our method is successfully applied to ${\mathrm{Cr}}_2{\mathrm{O}}_3$ and orthorhombic $R\mathrm{Mn}{\mathrm{O}}_3$ with linear and quadratic ME effects, respectively. We revealed all the possible microscopic origins of every nonzero ME coefficient that cannot be easily fulfilled by other theoretical methods. Moreover, the contribution from each mechanism can be located down to specific spins or spin pairs. Our theoretical approach is capable of providing a detailed guide on exploring rich spin-induced magnetoelectrics with strong ME effects.

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• Electric polarization
• Magnetic order
• Magnetoelectric effect
• Multiferroics